Extremal Functions for Singular Moser-Trudinger Embeddings

We study Moser-Trudinger type functionals in the presence of singular potentials. In particular we propose a proof of a singular Carleson-Chang type estimate by means of Onofri's inequality for the unit disk in $\mathbb{R}^2$. Moreover we consider Adimurthi-Druet type functionals on compact surfaces with conical singularities and discuss the existence of extremals for such functionals extending previous results by Cast\`o and Roy.